ANNOUNCEMENTS
MTH 434/534 — Winter 2024

3/23/24

Course grades have been submitted, but may not be visible until Monday.

There will be an opportunity to go over the exam next term.

3/22/24

Final exam scores have been published on Gradescope, but I'm still working on course grades

IF your grade were determined only by the raw score on the final exam, it would be:

• $\ge42$: A
• 32–41: B
• 28–31: B–
• 24–27: C
• 20–23: C–
• $\lt20$: F

Please treat the above ranges as a guide; scores close to the cutoffs could still go either way.

3/20/24

Below are some of the answers to the final:

2. $\frac1{u^2+v^2}\left(\frac{ \partial^2f}{\partial u^2} + \frac{\partial^2f}{\partial v^2} \right)$
3. $\omega^\theta{}_\phi=-\cos\theta\,d\phi=-\omega^\phi{}_\theta$ (all others zero)
4. $ds^2 = -\cosh(X)^2\,dT^2+(dX-\sinh(X)\,dT)^2 = -(dT+\sinh(X)\,dX)^2+\cosh(X)^2\,dX^2$
6. $-\frac1{a^2\cosh^4\left(\frac{z}{a}\right)}$

3/14/24

Happy Einstein's Birthday! And Happy Pi Day!

Notes corresponding to today's discussion of the torus can be found here.

A live curvature computation for the torus using Sage can be found here.

3/13/24

I will be in my office tomorrow starting no later than 1 PM.

As reported in class, I will not be available next week for in-person office hours. Email should reach me; Zoom may be possible.

Despite what Gradescope says, a score of 15 on HW 6 corresponds to 100%.

MTH 434 students could earn up to 5 points of extra credit; the maximum score reported by Gradescope includes this possibility.

3/12/24

Notes corresponding to today's class can be found here.

With apologies, the discussion of geodesic curvature in DFGGR is only correct up to sign.

Equation (18.64) can't be correct as written, since, as was pointed out in class, the LHS does not change sign if you go backward, but the RHS does change sign.

One way to resolve this issue is to define the sign of the geodesic curvature so that it is positive if the principal unit normal vector agrees with $\hat{e}_3\times\hat{T}$, but negative if these vectors are equal and opposite.

That usage is in fact implicit in (18.64), since $\frac{d\hat{T}}{ds}$ is known to be the principal unit normal vector, written as $\hat{N}$ in §18.1, but only equal to $\pm\hat{N}$ when using (18.63).

The Euler characteristic can be defined both for nonorientable surfaces and for surfaces with boundary; see for instance Wikipedia.

It is straightforward to generalize the Gauss–Bonnet Theorem to handle boundaries. (Do you see how?) However, it is more challenging to interpret integration over nonorientable surfaces.

3/10/24

The final exam is scheduled for Tuesday, 3/19/24, from 6–7:50 PM in Kidder 350.

There will be a review session during class on Thursday, 3/14/24.
Come prepared to ask questions.

A formula sheet will be available on the final. You can find a draft copy here.

• The exam is closed book.
• The exam will cover everything discussed in class through Tuesday, 3/12/24.
• The exam will cover material from the entire course, with an emphasis on new material.
• Important new topics are the connection ($\omega^i{}_j$) and curvature ($\Omega^i{}_j$).
• Important old topics are the exterior product ($\wedge$), Hodge dual ($*$), and exterior differentiation ($d$).

3/8/24

Two mathematicians are talking on the telephone. Both are in the continental United States. One is in a West Coast state, the other is in an East Coast state. They suddenly realize that the correct local time in both locations is the same! How is this possible?

Give up? Some hints can be found here.

3/7/24

Notes corresponding to today's class can be found here.

A live curvature computation using Sage for the (2-dimensional surface of the) sphere can be found here.

The pictures of the tractrix and pseudosphere shown at the beginning of class can be found here.

These pictures are also in my book on special relativity, an early version of which can be found here.
(The pseudosphere is discussed in §14.5.)

3/6/24

Notes corresponding to yesterday's class can be found here and here.
A brief summary of two examples we considered can be found here.

These notes also include a calculation of the connection in spherical coordinates, showing how to use the torsion-free condition.
(The Bianchi identity at the end will be covered in tomorrow's class.)

3/5/24

A live curvature computation using Sage, as discussed in class today, can be found here:

Euclidean 3-space in spherical coordinates

3/1/24

Notes corresponding to Thursday's class can be found here.

2/28/24

Here finally are the answers to the midterm questions, as discussed in class last Thursday.

1. (a) $0$ (b) $2\,dx\wedge dy\wedge dz$ (c) $0$ (d) $10\,dx\wedge dy\wedge dz$
2. (a) $0$ (b) $2\,dx\wedge dy\wedge dz\wedge dt$ (c) $0$ (d) $\alpha=x\,dy+z\,dt$ (or similar)
3. (a) $S=dx\wedge dy+dz\wedge dw$ (among others); $A=dx\wedge dy-dz\wedge dw$ (among others)
4. (a) $\frac{\partial^2 f}{\partial t^2} - \frac{1}{r}\frac{\partial}{\partial r} \left(r\frac{\partial f}{\partial r}\right) -\frac{1}{r^2} \frac{\partial^2 f}{\partial \phi^2}$
5. (a) $g(dh,dh) + h\triangle h$ (b) $\grad\cdot(h\grad h) = |\grad h|^2 + h\,\triangle h$
6. (a) $2\,\pi^2 a^4$

Should you have any questions about the midterm problems, you are strongly encouraged to try again on your own, then come to office hours, where you can also look at worked solutions.

2/27/24

Notes corresponding to today's class, including some material covered last Thursday, can be found here.

The moral of HW #X is not that Stokes' Theorem is subtle, but rather that orthonormal bases matter!

The easiest way to see that $\alpha=-r^2\cos\theta\,d\phi$ is not well-defined on the sphere is to express it as $\alpha=-(r\cot\theta)(r\sin\theta\,d\phi)$, whose component $\alpha_\phi=-r\cot\theta$ is badly behaved at the poles.
Alternatively, construct the vector field corresponding to $\alpha$, namely $\vf\alpha=-(r\cot\theta)\Hat\phi$, since $\alpha=\vf\alpha\cdot d\vf{r}$.

2/22/24

All exams were initially graded using the same rubric, then adjusted by class level.

• Raw exam scores were computed by adding together your base points and extra credit points for each problem, then dropping the problem with the lowest score.
• For MTH 434 students, adjusted scores were computed by rescaling your raw score on problem 2 by 10/7 (with a maximum of 12), then proceeding as above. Your adjusted score will be greater than or equal to your raw score.
• For MTH 534 students, adjusted scores were computed by rescaling your raw scores on problems 2, 3, and 5 to absorb the extra credit, then proceeding as above. Your adjusted score will be less than or equal to your raw score.

IF your grade were determined only by your adjusted midterm score (rounded to the nearest integer if necessary), it would be:

• $\ge44$: A
• 40–43: A–
• 37–39: B+
• 31–36: B
• 25–30: C
• $\lt25$: F

2/20/24

With apologies for the late announcement, I expect to be in my office this afternoon no later than 1:30 PM, and most likely by 1 PM.

2/19/24

HW #X will be accepted late without penalty until the start of class on Tuesday, 2/27.

HW #5 is due on Thursday, 2/29, and is more important than HW #X. Manage your time wisely!

2/18/24

Apart from computational errors, mostly minor, the most common error on HW #4 was:

Failing to express $\alpha$ in terms of an orthonormal basis: $\alpha = \sum_i\alpha_i h_i du^i$.

A sample solution for spherical coordinates can be found here.

2/17/24

As already announced, the first part of §6.1 in the text provides a good review of $\wedge$, $*$, and $d$.

An older document covering similar content can be found here.

2/16/24

Equation (15.87) on page 185 of DFGGR is incorrect: The numerator of each partial derivative should be $f$.

The wiki version here is correct.

A complete list of known typos in DFGGR is available here.

2/15/24

Notes corresponding to today's discussion of Maxwell's equations can be found here and here.

Notes from today's review can be found here

As mentioned at the beginning of class today, §6.1; provides a good overview of $\wedge$, $*$, and $d$.

As also mentioned at the beginning of class, (most of) the problems at the end of each chapter are a good review.

2/13/24

Notes covering part of today's class on integration can be found here.

The Sage code I demonstrated in today's class for computing $\wedge$, $*$, and $d$ in orthogonal coordinates can be found here.

This one is alpha software! It should work for the coordinate systems from HW #4, but is not guaranteed otherwise.

2/12/24

With apologies for the delay, HW #3 is almost has finally been graded.

There were however lots of sign errors...

See yesterday's announcement for one way to double-check your work.

To adapt the given Sage code for Minkowski 4-space, change the signature (sig=1), rename the coordinates, remove the factors of $r$ and $\sin\theta$ (so [1,1,1,-1] in the makeg command), and remove the assume command.

2/11/24

There are a variety of software packages capable of manipulating differential forms, including packages for both Maple and Mathematica. Another option is the open-source software SageMath, which is also available through a cloud server.

I have used most of these packages myself. Feel free to contact me for advice and assistance.

I have set up an experimental interface to Sage here, which should be fairly easy to adapt to other examples. Some tips:

• The operations $d$, ${*}$, and $\wedge$ should all work, entered as d(), star(), and Wedge(), respectively.
• Wedge can be used with more than two arguments.
• You can not add more boxes, but you can enter multiple lines of code in each box.
• Only the last result will be printed.
• You may need to use the Show() command to see the result you expect.
• Computations in one box can be used in later boxes.

This is beta software! Please do let me know if it does not work as expected.

2/9/24

Notes corresponding to yesterday's class can be found here.

The midterm is scheduled for Tuesday, 2/20/24 (Week 7).

• The exam will be a traditional, timed, closed-book exam during the regularly-scheduled class period.
  The exam will be graded only for content, not for presentation.
• A formula sheet will be available on the midterm. A draft is available here.
  Feel free to send me suggestions for additions.
• Part of Thursday's class next week (2/15) will be devoted to review.
  Come prepared to ask questions!

2/7/24

Notes corresponding to yesterday's class can be found here and here.

Formulas for divergence and curl (and gradient) in spherical and cylindrical (and rectangular) coordinates can be found here.

2/6/24

By popular request, today's assignment may be submitted until midnight tonight.
(Gradescope will mark it late after 4 PM, but I won't.)

Be warned that it is unlikely that a similar extension will be available next week.

2/2/24

The midterm will be either on Thursday 2/16 (Week 6) or Tuesday 2/20 (Week 7).

If you have any concerns about this timing, please let me know immediately.

The exam will be closed-book, during the regularly-scheduled class period.

A formula sheet will be provided on the midterm. A copy will be made available beforehand.

2/1/24

Notes from this week can be found here, here, and here.

You can find out more about the reasons we will use the "physics" convention for the names of the spherical coordinates in our paper: Spherical Coordinates, Tevian Dray and Corinne A. Manogue, College Math. J. 34, 168–169 (2003)

The short answer is that most nonmathematicians will likely need to switch conventions anyway...

1/27/24

Notes corresponding to Thursday's class can be found here and here.

Electronic notes from class can be found here.

Here is an explicit example of "Einstein summation":

Let $\alpha\in\bigwedge^1(\RR^2)$ be a 1-form in two dimensions, and let $A$ be the linear map that swaps $dx^1$ ($=dx$) and $dx^2$ ($=dy$). Determine the matrix $(a^i{}_j)$ of $A$ in this basis. Then determine the action of $A$ on 2-forms, and compare with $\det(A)$.

The general solution (for any $A$) is:

The components $(a^i{}_j)$ of $A$ are defined by $A(dx^i)=a^i{}_j\,dx^j$, where $i$ is fixed and there is a sum over $j$. So \begin{align} A(dx^1\wedge dx^2) &= A(dx^1)\wedge A(dx^2) = (a^1{}_i\,dx^i)\wedge (a^2{}_j\,dx^j) \\ &= ... = (a^1{}_1\,a^2{}_2 - a^1{}_2\,a^2{}_1) \,dx^1\wedge dx^2 = (\det A) \,dx^1\wedge dx^2 \end{align} where there is now a double sum over $i$ and $j$ in the third expression.

Make sure that you can follow these "index gymnastics", and that you can determine the components $a^i{}_j$ for the specific example given above.

1/23/24

Notes corresponding to today's class can be found here and here.

An annotated image showing $dx+dy$ can be found here.
The remaining pictures are in the text, and are also available here.

You can find more pictures of differential forms (including higher-rank forms) in Chapter 4 of the book Gravitation by Misner, Thorne, and Wheeler. A very interesting discussion of stacks and similar (but less standard) geometric interpretations of forms can be found in the book Geometrical Vectors by Weinreich.

Both books are available in the OSU library.

Another book worth looking at is Visual Differential Geometry and Forms by Needham.

I do not believe the OSU library has a copy of this book, but I have one in my office.

Geometric Algebra is based on differential forms and is an increasingly popular approach to describing rotations in computer science.

One place to get started is the book Geometric Algebra by Dorst, Fontijne, and Mann, whose website can be found here.

1/22/24

The $\LaTeX$ workshop scheduled for 1/18 (see announcement below dated 1/12) has been rescheduled for Thursday 1/25.

R 1/25 12–1:30 PM in Kidder 108J

1/20/24

If you're interested in finding out more about my current research, come to the physics colloquium on Monday:

What Kind of a Beast Is It?
The exceptional Lie Algebra $\mathfrak{e}_8$ and the Standard Model of Particle Physics
Corinne Manogue
M 1/22 at 4 PM in Weniger 149
(The colloquium will also be streamed via Zoom; if you want the link, ask me.)

1/19/24

Grades for HW #1 have been published on Gradescope (only).
I have posted one possible solution to the first homework assignment.
Can you spot the (minor) flaw in my logic?

If you don't get the score you were hoping for on this assignment, I encourage you to come to my office hours and/or to touch base with me via email about how things are going.

Here are some of the criteria that are used to assess your written work in this course:

Content:

• Correct logical reasoning;
• Correct computation;
• Correct answer to the question as posed;
• Assumptions clearly stated.

Presentation:

• Self-contained; should ideally be readable in 5 years without other sources;
• Clarity; the logical flow should be readily apparent;
• Typeset or carefully written;
• Good use of both inline and displayed equations;
• Figures are always a bonus.

1/18/24

THERE WILL BE NO CLASS TODAY, THURSDAY 1/18.

As you should already be aware, the OSU campus will remain closed today.

The first assignment has been graded and should be available on Gradescope later today.

The homework assignment originally due on Tuesday 1/23 is now due on Thursday 1/25.

You should be able to answer the first question based on material already covered in class.
We will discuss pictures of vector forms on Tuesday. If you want to start on the second question before then, read §13.8 in the text.

I will again hold office hours today via Zoom at the regularly scheduled time (2:30–3:30 PM).

I am again available during the regularly scheduled class time, but likely won't be on Zoom unless you contact me first via email.

1/17/24

The second assignment (HW #2) has been updated.

If your copy doesn't have an extra credit problem, download it again.

Today's power outage shut down the College of Science webserver, which hosts the course website.

I will therefore post duplicate copies of assignments in the Files section of the course Canvas site.

(There is no need to access the assignment on Canvas unless the power goes out again.)

1/16/24

With apologies, the link to my Gradescope information page appears to have been broken, but has now been fixed.

You are encouraged to read this page prior to submitting the assignment due today.

1/15/24

THERE WILL BE NO CLASS TOMORROW, TUESDAY 1/16.

As you should already be aware, the OSU campus will remain closed tomorrow.

The homework assignment due tomorrow should be submitted as planned; the due date and time remain unchanged.

Late submissions will be viewed sympathetically if they are due to weather-related issues.

You are reminded that uploading photographs to Gradescope is discouraged; please use a scanning app instead.
(See these instructions for further details.)

UPDATE: I will hold office hours tomorrow via Zoom at the regularly scheduled time (2:30–3:30 PM).

The Zoom link has just been posted as a Canvas announcement.

I am also available during the regularly scheduled class time, but likely won't be on Zoom unless you contact me first via email.

1/13/24

Here is a lightly edited list of the basic linear algebra topics that arose in class this week.

You should review these topics if you are rusty!

• vector space
• linear (in)dependence
• basis
• change of basis
• span
• linear transformation
• eigenvalues
• eigenvectors
• determinants

1/12/24

There was some minor confusion over the two assignments this week.

• HW #0 (brief introduction) was due yesterday via email.
• HW #1 (cross product) is due Tuesday, 1/16/23, by the start of class (4 PM) via Gradescope.

The homework page has been updated slightly to reflect the above numbering.

OSU's student chapter of the Society of Industrial and Applied Mathematics (SIAM) is sponsoring an introductory workshop on $\LaTeX$ from 12–1:30 PM on January 18 January 25 in Kidder 108J (the computer lab adjacent to the MSLC). Registration is not required.

From the announcement:

The Mathematics Department's Tyler Fara will guide a follow-along demonstration overviewing the basics of using a LaTeX editor and a variety of examples to help you get started typing.

1/11/24

A list of derivative rules in differential notation is available here.

Notes roughly corresponding to today's class can be found here.

We haven't yet fully discussed the derivatives shown on the second page.

1/10/24

The cap should have been raised today so as to allow all students on the waitlist to register.

If you are unable to register, please let me know, and please do make sure to get on the waitlist.

1/1/24

All assignments will be posted only on the homework page.

Assignments will not be posted on Canvas.

All assignments should be submitted via Gradescope.

Further information can be found on my own information page for Gradescope.

12/4/23

The text can be read online as an ebook through the OSU library.

There is also a freely accessible wiki version available, which is however not quite the same as the published version.

A schedule from a previous year can be found here.

This schedule is a good approximation to what we will cover when.